On the lack of selection for the transport equation over a dense set of vector fields

2024/04/30 by Jules Pitcho, Pitcho, Jules
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #advanced mathematical theories #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2404.19687

Abstract

We construct a set of bounded vector fields dense in L^p((0,2);W^s,p_loc(ℝ²;ℝ²)) for 1≤ p <+∞ and 0≤ s<1 with p<1/s for which smooth regularisation of the vector field does not give a selection criterion for the continuity equation, thereby showing that the two examples constructed in [Calc. Var. Partial Differ. Equ. 59 (2019), Ciampa, Crippa and Spirito] and [Ann. Math. Qué. 46 (2022), De Lellis and Giri] are generic.

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